0.11 Lecture 12:interconnected systems and feedback  (Page 3/5)

$Y(s)/X(s)\Rightarrow 1/\beta (s)\begin{array}{cc}& \end{array}\text{and}\begin{array}{cc}& \end{array}Y(s)/D(s)\Rightarrow 0$

4/ Improve dynamic characteristics

Our model of an op-amp was simplistic. A more realistic model that takes the finite bandwidth of the op-amp into account is shown below.

The finite bandwidth of the op-amp is modeled as

$K(s)=\frac{K}{s+\alpha }$

Using Black’s formula we obtain

$H(s)=\frac{K(s)}{1+\mathrm{\beta K}(s)}=\frac{\frac{K}{s+\alpha }}{1+\beta \frac{K}{s+\alpha }}=\frac{K}{s+(\alpha +\mathrm{\beta K})}$

Therefore, the frequency response is

$H(\mathrm{j\omega })=\frac{K/(\alpha +\mathrm{\beta K})}{\mathrm{j\omega }/(\alpha +\mathrm{\beta K})+1}$

the impulse response is

$h(t)={\text{Ke}}^{-(\alpha +\mathrm{\beta K})t}u(t)$

and the step response is

$s(t)=\frac{K}{\alpha +\mathrm{\beta K}}(1-e^{-(\alpha +\mathrm{\beta K})t})u(t)$

We compare the open-loop with the closed-loop characteristics. We use typical parameters of a model 741 op-amp, $\text{K}=\text{8}\times {\text{10}}^6$ and α = 40 rad/sec.

Effect on pole-zero diagram

As the loop gain is increased from 0 the pole moves out along the negative real axis from −40 rad/s and when β = 0.1 the pole reaches $-8\times {\text{10}}^5\text{rad}/s$ . This type of diagram which shows the trajectory of the closed loop poles in the complex s-plane as the gain is changed is called a root locus plot.

Effect of feedback on frequency response.

B

Effect of feedback on step response.

As the loop gain is increased from 0 the time constant of the step response decreases, i.e., the system responds faster.

Summary of effect of feedback on dynamic characteristics.

5/ Reduce the effect of nonlinear distortion

A common power amplifier configuration found in many electronic systems (e.g., stereo amplifiers) is the push-pull emitter follower amplifier.

This configuration has an inherent nonlinearity. When $v_i>\text{0}\text{.}\text{6 V}$ , T1 conducts and when , T2 conducts. Thus, there is a dead zone between −0.6 and +0.6 V where neither transistor conducts.

The transfer characteristic, which contains an idealized dead zone, is shown below.

In an audio amplifier, the type of distortion caused by the dead zone is called crossover distortion.

We examine the use of feedback to reduce cross-over distortion with the aid of MATLAB’s block diagram language SIMULINK.

6/ Effect of cross-over distortion on simple signals and on music

The figure shows the effect of crossover distortion on a sinusoid obtained with Simulink. A demo will show the effect of cross-over distortion in an audio amplifier on simple signals and on music and the role of feedback to minimize this distortion.

IV. DYNAMIC PERFORMANCE OF FEEDBACK SYSTEMS

1/ Simple position control system with proportional controller

The objective of the position control system shown below is for the output position Y (s) to track the input signal X(s).

We can use Black’s formula to find H(s) as follows

$H(s)=\frac{\frac{K}{(s+1)(s+\text{100})}}{1+\frac{K}{(s+1)(s+\text{100})}}=\frac{K}{(s+1)(s+\text{100})+K}$

How good is this system at controlling the motor position?

Let us examine the steady-state step response first. The system function is

$H(s)=\frac{K}{(s+1)(s+\text{100})+K}$

The steady-state response to a unit step is simply the response to the complex exponential $x(t)=\text{1}\text{.}{e}^{0\text{.}t}=\text{1}$ which is y(t) = 1.H(0) $e^{0\text{.}t}=\text{K/}(\text{100}+K)$ . The position error ε=K/(100+K)−1=−100/(100+K). Hence, this position control system (with proportional controller) has an error that can be made arbitrarily small, although not zero, as the gain is made arbitrarily large.